Diophantine sets are subsets defined by polynomial equations over integers. More precisely We obtain them basically by projecting integers solutions of some polynomial equations. Observation 1: Using the statement iff we see that there is no difference between one polynomial equation and a system of polynomial equations. Example: Even numbers are Diophantine: Odd numbers: Non-negative […]
Category: Number Theory
Mordell’s Equation
The Diophantine equation is called the Mordell equation. Here is a fixed nonzero integer, and we ask for integer or rational solutions . This equation is simple to write down, but it already contains much of the arithmetic complexity of elliptic curves. For , the curve is nonsingular. Indeed, the right-hand side has no repeated […]
Sophie Germain’s Theorem
As early attempts to the proof of Fermat’s last theorem, many mathematicians solved the problem for small exponents. While these special cases are being studied, Sophie Germain, a French mathematician, came up with the following interesting result. (Look at https://www.agnesscott.edu/lriddle/women/germain.htm for her fascinating and revolutionary story) Theorem 1: For any odd prime such that is […]
Kürschâk and Nagel’s theorems (Erdos 1932)
Consider the familiar reciprocal sums None of the above quantities are integers.The first, second, and fourth cases all follow from one very elementary principle. One looks for a prime which occurs in one denominator more strongly than it occurs in every other denominator. After the fractions are put over a common denominator, every term except […]
Betrand Postulate : Erdos( 1932)
Bertrand’s postulate states that for every integer , there is a prime satisfying The statement is elementary, but it is remarkably strong: no matter how far one goes along the number line, one never encounters a multiplicative gap as large as a factor of containing no primes. Erdős’s proof (1932) of this fact is centered […]
Zeta(2)
Basel Problem (1644) asks to find the exact value of the series Euler (1735) showed that At first glance, this is a problem about a list of numbers. Yet its answer, contains , a constant associated with circles, periodicity, and geometry. The surprise is not merely that the sum has a closed form. It is […]
Eisenstein’s Lattice Point Proof of Quadratic Reciprocity
The law of quadratic reciprocity is one of the central results of classical number theory. Gauss called it the Theorema Aureum, or Golden Theorem. Given two distinct odd primes and , it relates the question of whether is a square modulo to whether is a square modulo . These are questions about different moduli, yet […]
